On scaling limits of planar maps with stable face-degrees

We discuss the asymptotic behaviour of random critical Boltzmann planar maps in which the degree of a typical face belongs to the domain of attraction of a stable law with index α ∈ (1,2]. We prove that when conditioning such maps to have n vertices, or n edges, or n faces, the vertex-set endowed with the graph distance suitably rescaled and the uniform probability measure converges in distribution in the so-called Gromov-Hausdorff-Prokhorov topology towards the celebrated Brownian map when α = 2, and, after extraction of a subsequence, towards another 'α-stable map' when α < 2, which improves on a first result due to Le Gall and Miermont who assumed slightly more regularity.